On singular integrals of Calderón-type in $\mathbb R^n$, and BMO

Abstract

We prove $L^p$ (and weighted $L^p$) bounds for singular integrals of the form

where $E(t) =$ cos $t$ if $\Omega$ is odd, and $E(t) =$ sin $t$ if $\Omega$ is even, and where $\bigtriangledown A \in$ BMO. Even in the case that $\Omega$ is smooth, the theory of singular integrals with "rough" kernels plays a key role in the proof. By standard techniques, the trigonometric function $E$ can then be replaced by a large class of smooth functions $F$. Some related operators are also considered. As a further application, we prove a compactness result for certain layer potentials.